Fourier Series in Fractional Dimensional Space

TL;DR

This paper introduces a fractional Fourier series in fractional dimensional space for periodic functions, extending classical Fourier analysis through basis transformation and fractional derivatives, enabling new analytical tools.

Contribution

It presents a novel fractional Fourier series framework in fractional dimensional space, including basis function extension, coefficient calculation, and fractional derivative representation.

Findings

Defined fractional Fourier series of arbitrary order α

Derived coefficients for fractional Fourier series

Showed fractional derivatives can be represented via fractional Fourier series

Abstract

In this paper, a Fourier series in fractional dimensional space is introduced for an arbitrarily periodic function $f(t;\alpha)$. We call it fractional Fourier series of the order $\alpha$. Extending the basis functions of the linear space into fractional one, by rotation transformation, we define a real and complex Fourier series and obtain their coefficients. It is also shown that the fractional derivative of a periodic function can be realized through (fractional) Fourier series with modified coefficients.